### Abstract

It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv^{2}(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

Original language | English (US) |
---|---|

Pages (from-to) | 2603-2611 |

Number of pages | 9 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Block coloring
- Chromatic index
- Parallel class
- Steiner triple system

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(4), 2603-2611. https://doi.org/10.1137/17M1114338

**Steiner triple systems with high chromatic index.** / Bryant, Darryn; Colbourn, Charles; Horsley, Daniel; Wanless, Ian M.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 31, no. 4, pp. 2603-2611. https://doi.org/10.1137/17M1114338

}

TY - JOUR

T1 - Steiner triple systems with high chromatic index

AU - Bryant, Darryn

AU - Colbourn, Charles

AU - Horsley, Daniel

AU - Wanless, Ian M.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

AB - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.

KW - Block coloring

KW - Chromatic index

KW - Parallel class

KW - Steiner triple system

UR - http://www.scopus.com/inward/record.url?scp=85040375717&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040375717&partnerID=8YFLogxK

U2 - 10.1137/17M1114338

DO - 10.1137/17M1114338

M3 - Article

VL - 31

SP - 2603

EP - 2611

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -